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Unformatted text preview: e following tableau was developed by Hammer and Hollingsworth 18]. 1 2 1 2 ; + p 3 6 p 3 6 1 4 1 4 + 1 4 p 3 6 1 2 ; 1 4 1 2 p 3 6 This method is derived in Gear 15], Section 2.5. Example 3.3.3. Let us examine the region of absolute stability of the implicit midpoint rule (3.3.2). Thus, applying (3.3.2) to the test equation (3.2.8) we nd Y1 = yn 1 + h2 Y1 ; and yn = yn 1 + h Y1: ; Solving for Y1 y Y1 = 1 ;nh 1=2 ; and eliminating it in order to explicitly determine yn as 1 yn = 1 + 1 ;hh =2 yn 1 = 1 + h =2 yn 1: ; h =2 ; ; Thus, the region of absolute stability is interior to the curve 1 + z=2 = ei 1 ; z=2 Solving for z z=h : i =2 e ei z = 2 1 ; e i = ; 2 e i =2 ; e 1+ e+ i =2 i =2 ; ; 20 = ;2i tan =2: Im(h λ) Re(h λ) Figure 3.3.1: Region of absolute stability for the implicit midpoint rule (3.3.2). Since z is imaginary, the implicit midpoint rule is absolutely stable in the entire negative half of the complex z plane (Figure 3.3.1). Let us generalize the absolute stability analysis presented in Example 3.3.3 before considering additional methods. This analysis will be helpful since we will be interested in developing methods with very large regions of absolute s...
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